I've been waiting for this one for a while. Ghost Trick: Phantom Detective is a title that won accolades on the Nintendo DS last year, and it's now arrived on iOS in a universal version (that's free to download to boot). The idea is that you're a ghostly "phantom detective" who must solve his own murder. You can't directly interact with the living, but you can mess with inanimate objects in their world. So move objects around and try to change the fate of other dead people. It sounds morbid, but there's a fun sense of humor and style that keep it entertaining.
The first two chapters are free to play through, and you can buy the rest for US$9.99 total, or $4.99 in three different chapter packs. Even for the full price, that's cheaper than you'll find it for sale on the DS. There's full Game Center integration and the build is universal, which makes this one of the best ports that Capcom has ever released. It's great to see such a popular game get a decent port to the App Store. Definitely pick it up and try it out. You might find yourself compelled enough to buy the whole and finish the story.